The transformation optics technique is employed in this paper to design two optical devices — a two-dimensional polarization splitter and a three-dimensional polarization rotator for propagating beams. The polarization splitter translates the TM- and the TE-polarized components of an incident beam in opposite directions (i.e., shifted up or shifted down). The polarization rotator rotates the polarization state of an incoming beam by an arbitrary angle. Both optical devices are reflectionless at the entry and exit interfaces. Design details and full-wave simulation results are provided.
©2008 Optical Society of America
Given a spatial coordinate transformation, the transformation electromagnetics/optics technique [1, 2] is based on the observation that Maxwell’s equations written in the transformed system can be interpreted as inhomogeneous, anisotropic compression and expansion of the material tensor parameters of the original space. The coordinate transformation technique has led to numerous device designs that perform functions which have not been possible or difficult to achieve using conventional design methodologies, with the most notable example being the invisibility cloaks [3, 4, 5, 6, 7, 8, 9, 10, 11]. A field-rotating device  and a circular field concentrator  have also been designed using transformation optics. Recently, an embedded coordinate transformation approach  was introduced, where a discontinuity at the outer boundary of the transformed spatial domain is permitted. Beam shifter/splitter designs were presented and demonstrated in two dimensions  based on this technique. The embedded transformation approach also has led to additional optical device designs such as wave collimators or cylindrical-to-planar wave converters [14, 15, 16], beam bends [17, 16], and beam compressors/expanders .
In this paper, a beam splitter based on the polarization state of the incident field is presented in two dimensions. This is unlike the beam splitter in , where an incident beam was split into two beams of the same polarization having narrower widths. In addition, a three-dimensional (3D) reflectionless polarization rotator is demonstrated. Material parameter specifications are derived and the full-wave electromagnetic simulation results are presented for verification.
2. Polarization splitter
To design a polarization splitter, consider the embedded transformation illustrated in Fig. 1. The mapping is from the original (x′,y′, z′) space to the transformed (x,y, z) space where only the transformed space is shown. The original space is a two-dimensional (2D) cylinder of free space with a rectangular cross section of the size 2w×2h (b 0=c 0=-h, b 3=c 3=h).
The device operates on 2D beam illuminations from the region x<-w. One observes that the total fields can be described as a superposition of the decoupled set of TE and TM modes. The TE-mode fields interact only with the material parameters εzz, µxx, µxy, µyx, and µyy, whereas the TM-mode fields are affected only by µzz, εxx, εxy, εyx, and εyy. Therefore, the former set of material parameters can be designed to manipulate the TE-mode fields and the latter set to control the TM-mode fields, completely independent of each other. Hence, the polarization splitter design is obtained by shifting a TE-mode incident field in the +ŷ direction and a TM-mode field in the -ŷ direction. The transformations illustrated in Fig. 1 achieve this function, similar to those introduced in . In region i (i=1,2,3), the transformed coordinates (x,y, z) are given by
The function y=yi(x) (i=0,1,2,3) represents a straight line segment connecting (-w,bi) and (w,ci) as illustrated in Fig. 1(a). The rectangular cylindrical region in the original system specified by -w≤x′≤w, b 1≤y′≤b 2 is transformed into the cylindrical region of the parallelogram cross section bounded by -w≤x≤w, y 1(x)≤y≤y 2(x). The regions 1 and 3 are either contracted or expanded to make the total transformed volume of the device to coincide with the original volume. The width of region 2 is kept constant at b 2-b 1=c 2-c 1 along the shift in the ±ŷ directions in Fig. 1. This constant width leads to a reflectionless property at the exit interface at x=w. The distance of the vertical beam shift in the ŷ direction is equal to c 1-b 1=c 2-b 2.
for region i (i=1,2, 3). Hence, only the material parameters εzz, µxx, µxy, µyx, and µyy are determined based on one set of values for c 1 and c 2 from 
On the other hand, the material parameters µzz, εxx, εxy, εyx, and εyy are based on a different set of values for c 1 and c 2.
The performance of a design example under a Gaussian beam illumination propagating in the +x̂ direction is analyzed using a full-wave analysis tool based on the finite-element method (COMSOL Multiphysics), where the simulation results are shown in Fig. 2. The dimensions of the splitter design is specified by w=0.1 m and h=0.3 m. The values of b 1 and b 2 for both polarizations are given by b 2=-b 1=0.15 m. Contours in black indicate the region boundaries for the two polarizations defined in Fig. 1. In order to shift up the TE mode and shift down the TM mode, the values of c 1 and c 2 were chosen as c 1=0, c 2=0.3 m for the TE mode and c 1=-0.3 m, c 2=0 for the TM mode. The incident Gaussian beam has a minimum waist of 0.05 m at the origin and it illuminates the splitter normally at 3 GHz. Figure 2(a) shows a snapshot of the total electric field when a TE-mode beam is incident upon the splitter. It is seen that the beam is completely shifted in the +ŷ direction. In addition, the device is reflectionless at both the entry and the exit boundaries. The same design shifts a TM-mode beam in the - ŷ direction, as demonstrated in Fig. 2(b), where the total ẑ-directed magnetic field distribution is shown. When a circularly polarized beam illuminates the splitter, the TE and TM mode components of the incident field are shifted in opposite directions and the two polarizations are completely separated at the exit plane. This is demonstrated in Fig. 3, where the magnitude of the Poynting vector is plotted. It can be clearly observed that the incident beam is split into two beams of the same widths, each carrying half the incident power.
The reflectionless property of the splitter design can be confirmed using the metric matching criteria proposed in . Let a set of orthonormal vectors in the original system be defined by
Their transformed versions (q1,q2,q3) are found using Eq. (4) as
Since we chose b 2-b 1=c 2-c 1=y 2(x)-y 1(x), it follows that a 22=1 and thus
Therefore, the metric matching conditions are satisfied at both interfaces.
An incident beam of arbitrary polarization will be separated into two linearly polarized beams having orthogonal polarization vectors with respect to each other at the output plane and beyond by the polarization splitter. Fig. 3 illustrates splitting a circularly-polarized incident beam, which has a 90° phase difference between the TM and the TE components of equal strengths. If these incident field components were in phase, a linearly polarized incident beam with the polarization vector (ŷ+ẑ)/√2 would result (simply obtained by adding the two incident beams illustrated in Fig. 2), but the magnitude plot of the Poynting vector will be exactly the same as in Fig. 3. If the strengths of the incidentTM and TE modes are different, either a linearly-polarized or an elliptically-polarized incident beam will form depending on the phase difference and the separated linearly polarized output beams will have unequal magnitudes.
3. Polarization rotator
Next, a 3D polarization rotator design is introduced. Consider the circularly cylindrical device of length 2l and radius a illustrated in Fig. 4(a). For a beam field propagating along the z axis in the +ẑ direction, it is desired that the polarization vector of the electric field be rotated by an angle α in the + ′ direction at z=l (the exit boundary) from the direction it would have if it propagated in free space. This can be achieved by continuously twisting the original cylindrical volume in the + direction as a function of increasing z′. One such a mapping from the original circular cylindrical system (ρ′,ϕ′, z′) to the transformed system (ρ,ϕ, z) is given by
The transformed system is illustrated in Fig. 4(b). Although the transformation is applied for the cylindrical volume, only the cylindrical shell at ρ′=ρ=a is shown for clarity. It is observed that the constant-ϕ′ line segments in the original system are mapped into helices in the transformed system. The material tensor parameters are found from Eq. (7) as
and ερϕ=εϕρ=ερz=εzρ=0. The parameters are neither functions of ϕ nor z. Moreover, ερρ and εzz are constants, while the other parameters depend only on ρ. When α=0, the material parameters reduce to those of free space.
An example rotator design was analyzed using COMSOL Multiphysics. The dimensions of the rotator are given by a=0.15mand l=0.025 m. The angle of rotation is set to α=π/2. The time-harmonic frequency of the incident beam is 6 GHz, such that the diameter is 6λ and the length is λ in terms of the free space wavelength. A circular cylindrical volume of free space with the thickness of 0.025 m was added on each side of the rotator. Finally, thin perfectly matched layers were placed on top of the air volumes. An x̂-polarized Gaussian beam with a minimum waist of 0.05 m at the coordinate origin illuminates the device at normal incidence from the -ẑ direction.
Figure 5 shows snapshots of the x̂ and the ŷ components of the electric field E=x̂Ex+ŷEy in the x-z and the y-z planes. At the entry plane corresponding to z=-l, E is linearly polarized in the x̂ direction. Within the rotator device (-l≤z≤l), E has both non-zero ̂x and ŷ components. However, it can be observed that only Ey is present past the exit boundary at z=l. Therefore, the polarization vector of the incident beam is completely rotated through 90° by the device. As a further verification, snapshots of the electric field components on the z axis are plotted with respect to z in Fig. 6. To test the performance of the polarization conversion, the fields are compared with the case of the same x̂-polarized Gaussian beam propagating through free space in the absence of the device (shown as dashed lines). In the range z<-0.025 m, the total field and the incident field are the same, which signifies that the device is reflectionless at both entry and exit boundaries. The envelope of Ex diminishes with increasing z as shown in Fig. 6(a). In comparison, the envelope of Ey grows at the same time within the device and the field is linearly polarized in the ŷ direction upon exiting. As expected, we see that Ey in the region z>0.025 m is the same as that of Ex for the case where the incident beam passed through an equivalent volume of free space.
The metric matching can be checked for the polarization rotator. First, the unit basis vectors in the original system are defined as
Therefore, the metric matching criteria Eq. (10) are satisfied for the polarization rotator and the device is expected to be reflectionless. It should be noted that the conditions are satisfied when the unit vectors q̂′ 1, q̂′ 2, and q̂′ 3 are evaluated at the transformed location.
Employing the coordinate transformation technique, two device designs that operate on the polarization of an incident beam were presented. By controlling the two sets of material parameters associated with the TE and the TM modes, the design for a 2D polarization splitter was obtained. Moreover, a 3D polarization rotator was designed by applying a transformation that amounts to twisting a circularly cylindrical volume around in the direction of beam propagation. The optical functions of the two devices were confirmed using full-wave simulations and the reflectionless properties were also verified.
This work was supported in part by the Penn State Materials Research Institute and the Penn State MRSEC under NSF grant DMR-0820404, and also in part by ARO-MURI award 50342-PH-MUR.
References and links
3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]
4. G.W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248 (2006). [CrossRef]
5. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006). [CrossRef]
6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. 1, 224–227 (2007). [CrossRef]
8. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007). [CrossRef]
9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008). [CrossRef]
10. D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]
11. Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008). [CrossRef]
12. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
13. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008). [CrossRef] [PubMed]
14. J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” Prog. Electromagn. Res. 81, 437–446 (2008). [CrossRef]
15. W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded coordinate transformation,” Appl. Phys. Lett. 92, 261903 (2008). [CrossRef]
16. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses, and right-angle bends,” New J. Phys. (to be published).